Measurement with a magnetic field

ABSTRACT

Material properties such as stress in a ferromagnetic material may be measured using an electromagnetic probe. While generating al alternating magnetic field in the object, and sensing the resulting magnetic field with a sensor, the signals from the magnetic sensor may be resolved into in-phase and guadrature components. The signals are affected by both geomaterical parameters such as lift-off any by material properties, but these influences may be separated by mapping the in-phase and quadrature components directly into material property and lift-off components, and hence a material property and/or the lift-off may be determined. The mapping may be represented in the impedance plane as two sets of contours representing signal variation with lift-off (A) (for different values of stress) and signal variation with stress (B) (for different values of liftoff), the contours of both sets (A, B) being curved. The stress contours (B) at a constant angle. Hence calibration measurements taken along a few contours of each set enable.

This invention relates to a method and apparatus in which anelectromagnetic probe is used to measure material properties in aferromagnetic material, for example stress, or to measure the separationof the probe from the surface of such a material.

The stresses in structures such as rails, bridges and pipelines, complexmechanisms such as vehicles and machinery, or simple devices such asstruts, cables or bearings arise from various causes including changesof temperature, and the loads and pressures due to use. There may alsobe residual stresses arising from the fabrication of the structure ordevice, and any bending that the structure or device was subjected toduring construction; the residual stresses arising from fabrication willalso be affected by any stress-relieving heat treatment. In somesituations (such as pipelines) the principal stress directions can beexpected to be in particular directions (circumferential andlongitudinal), whereas in other situations the stress directions arealso unknown. A variety of magnetic techniques are known to have somesensitivity to stress, although magnetic measurements are usually alsoaffected by other material properties such as microstructure. A way ofmeasuring stress in a steel plate is described in GB 2 278 450, thismethod using a probe containing an electromagnetic core to generate analternating magnetic field in the plate, and then combining measurementsfrom two sensors, one being a measure of stress-induced magneticanisotropy, and the other being a measure of directional effectivepermeability (DEP). The probe is gradually turned around so the magneticfield has a plurality of different orientations in the plate, and thesemeasurements are taken at each such orientation. The DEP signals areaffected not only by stress, but also by the lift-off from the surface(i.e. the gap or separation between the probe and the surface), and somust be corrected for lift-off.

According to the present invention there is provided an apparatuscomprises at least one probe for measuring material properties in anobject of ferromagnetic material and/or measuring the lift-off of theprobe from the surface of the object, the or each probe comprising anelectromagnet means, means to generate an alternating magnetic field inthe electromagnet means and consequently in the object, and a magneticsensor arranged to sense a magnetic field due to the electromagnetmeans; analysis means for resolving signals from the magnetic sensorinto a first component and a second component which are orthogonal inphase; means for mapping the first and second components directly intomaterial property and lift-off components; and means for deducing amaterial property from the material property component so determined,and/or a lift-off value from the lift-off component so determined.

The mapping requires a preliminary calibration, with a specimen of thematerial, to determine how the first and second components of the signalvary with lift-off (at a constant stress) and vary with stress (at aconstant lift-off), and deducing from the calibration measurements theapplicable mapping for any stress and any lift-off. The signals from thesensor are at the frequency of the alternating field, and the componentsmay be the components in phase with the current supplied to theelectromagnet means, and the component in quadrature to that. Themapping may be represented in the impedance plane (i.e. on a graph ofquadrature component against in-phase component) as two sets of contoursrepresenting signal variation with lift-off (for different values ofstress) and signal variation with stress (for different values oflift-off), the contours of both sets being curved. The contours or linesof one set intersect those of the other set at non-orthogonal angles.The angles at which the contours for constant lift-off (varying stress)intersect any one contour for constant stress (varying lift-off) areconstant along that contour. However the angles of intersection ofdifferent lift-off lines with any one stress line are different: theangle of intersection at a fixed lift-off varies slightly with stress.Hence measurements taken along a few contours of each set enable thepositions of the other contours of each set to be determined.

Surprisingly this simple mapping has been found to give an accuraterepresentation of the variation of the signals with material property(e.g. stress or microstructure); more surprisingly it enables thesevariations to be distinguished unambiguously from variations arisingfrom lift-off or other geometrical variations such as surface texture orcurvature. It may perhaps be presumed that material property changescause changes in the permeability of the steel in the magnetic circuitand these have both inductive and resistive (lossy) components, whereasgeometrical changes such as lift-off change the amount of air in themagnetic circuit, in which energy dissipation by eddy currents cannotoccur, so this of itself would by purely inductive (non-lossy). Wherethe object has a coating of a non-ferromagnetic material (such as apaint, or a ceramic), this coating will separate the probe from thesurface, so the lift-off measurement may indicate its thickness.

Preferably the electromagnet means comprises an electromagnetic core andtwo spaced apart electromagnetic poles, and the magnetic sensor ispreferably arranged to sense the reluctance of that part of the magneticcircuit between the poles of the electromagnet means. It is alsodesirable to arrange for such measurements to be taken with a pluralityof different orientations of the magnetic field, at a single location onthe object. This may be achieved using a single probe that is rotated atthat location, measurements being taken with different orientations ofthe probe, or using an array of probes of different orientations thatare successively moved to that location. In either case, the sensor orsensors provide a measure of the permeability of the material throughwhich the flux passes between the poles, and so provide a signalindicative of the effective permeability of the material; thecorresponding measurements at different probe orientations at a locationon the object hence indicate the effective permeability in differentdirections. The signals from this sensor may be referred to as areluctance, or ‘flux linkage’, signal.

The probe, or at least some of the probes, may also include a secondmagnetic sensor between the two poles and arranged to sense magneticflux density perpendicular to the direction of the free space magneticfield between the poles. This second sensor would detect no signal ifthe material were exactly isotropic; however stress induces anisotropyinto the magnetic properties of the material, and so the signalsreceived by the second sensor are a measure of this stress-inducedmagnetic anisotropy or ‘flux rotation’. The variations in the fluxrotation signals at different probe orientations, at a location on theobject, enable the directions of the principal stress axes to beaccurately determined. The flux rotation signals can also be related tothe stress.

The flux linkage signal from the or each probe is preferably backed-off,i.e. processed by first subtracting a signal equal to the signal fromthat sensor with the probe adjacent to a stress-free location. Thebacked-off signal is then amplified so the small changes in the fluxlinkage signal due to stress are easier to detect. This backing off isperformed after resolving into in-phase and quadrature components butbefore performing the mapping. Preferably the signals from the or eachprobe are digitized initially, and the backing-off and resolution areperformed by analysis of the digital signals.

To achieve penetration below the surface of the ferromagnetic object itis desirable to operate at alternating frequencies less than 200 Hz, forexample between 5 Hz and 100 Hz (which in mild steel providepenetrations of about 5 mm and 1 mm respectively). In other situations,where such depth of penetration is not required, higher frequencies canbe used, for example up to 150 kHz for a penetration of only about 15μm. The depth of penetration may be represented by the skin depth,δ=1/{square root}{square root over (()}πμ_(o)μ_(r)fk), where μ_(o) isthe permeability of free space, μ_(r) is the relative permeability ofthe material, k is its electrical conductivity, and f is the frequency.The frequency should be such that the skin depth is much less than thethickness of the object.

Generally, the more different probe orientations are used for takingmeasurements the more accurate the determination of stress levels andprincipal axes can be. In many cases the principal stress axes can beassumed to be aligned in particular directions—axial and circumferentialdirections in the case of a pipe, for example—so that the signal maximafor flux linkage signals would be expected to be along these directions,and the signal maximum for flux rotation signals would be along thebisection angles between these directions.

The probe, or at least some of the probes, may also include a thirdmagnetic sensor (a ‘flux leakage’ sensor) between the poles and arrangedto sense magnetic flux density parallel to the free space magneticfield. This third sensor will detect any flux leakage, this beinginfluenced by changes in material properties, lift-off, and cracks. Aswith the flux-linkage sensor, measurements at a location are preferablymade at different probe orientations.

The invention will now be further and more particularly described, byway of example only, and with reference to the accompanying drawings, inwhich:

FIG. 1 shows a diagrammatic view of an apparatus for measuring stress orlift-off;

FIG. 2 shows a longitudinal sectional view of a probe for use in theapparatus of FIG. 1;

FIG. 3 shows graphically the variation of the backed-off quadrature andin-phase components of the flux linkage signal with variations oflift-off, and with variations of stress.

Referring to FIG. 1, an apparatus 10 for measuring stress and/orlift-off includes a sensor probe 12 comprising sensors for both fluxlinkage and flux rotation, the probe 12 being attached to an electricmotor 14 which can be held by an operator, so the motor 14 can turn theprobe 12 with one end adjacent to a surface of a steel object 16 inwhich the stress is to be determined. The sensor probe 12 and motor 14are connected by a 2 m long umbilical cable 17 to a signalconditioning/probe driver unit 18. The unit 18 is connected by a longumbilical cable 19 (which may for example be up to 300 m long) to aninterface unit within a microcomputer 20, which has a keyboard 21.Operation of the apparatus 10 is controlled by software in themicrocomputer 20.

The interface unit within the microcomputer 20 generates sine and cosinefunctions at an angular frequency selectable by software, and buffersthe sine waveform for transmission to the unit 18 for driving the probe12. The amplitude of the transmitted waveform is also selectable bysoftware. It also provides signals to control the motor 14 and hence theangular position of the probe 12. The interface unit also providescontrol signals to the unit 18 to select which of the signals availablefrom the probe 12 is to the transmitted for analysis. It demodulates theselected input signal (flux linkage or flux rotation) to derive itsin-phase and quadrature components, filters the demodulated signal toremove high frequency components and to reduce noise, and converts theanalogue signals to digital form for input to the computer 20. It alsodetects the angular position of the probe 12 from signals provided by aposition encoder (not shown) on the motor 14.

The long umbilical cable 19 incorporates a coaxial cable to transmit theselected signal, and wires to control which signal is selected, tocontrol the motor 14, to transmit signals from the position encoder, totransmit the sinusoidal waveform, and to convey electrical power. Theunit 18 converts the drive waveform from a voltage to a current drivefor the probe 12; buffers and amplifies the signals from the probe 12;and selects which signal is to be transmitted to the microcomputer 20.It also buffers the signals from the position encoder for transmission,and drives the motor 14 in response to control signals.

Referring now to FIG. 2, the probe 12 is shown detached from the motor14, in longitudinal section although with the internal components shownin elevation (the connecting wires within the probe 12 are not shown).

The probe 12 comprises a cylindrical brass casing 24 of externaldiameter 16.5 mm and of overall height 60 mm, the upper half being ofreduced diameter whereby the probe 12 is attached to the motor 14. Theupper half of the casing 24 encloses a head amplifier 25. The lower halfencloses a U-core 26 of laminated mu-metal (a high permeabilitynickel/iron/copper alloy) whose poles 28 are separated by a gap 7.5 mmwide, and are each of width 2.5 mm, and of thickness 10 mm (out of theplane of the figure). The poles 28 are in the plane of the lower end ofthe casing 24, which is open. Around the upper end of the U-core 26 is aformer on which are wound two superimposed coils 30. One coil 30 a(which has 200 turns) is supplied with the sinusoidal drive current fromthe unit 18; the other coil 30 b (which has 70 turns) provides fluxlinkage signals. Between the two poles 28 is a former on which is wounda 1670-turn rectangular coil 32, about 4 mm high and 6 mm wide, and 6mm-square as seen from below, the windings lying parallel to the planeof the figure so the longitudinal axis of the coil 32 is perpendicularto the line between the centres of the poles 28. The coil 32 issupported by a support plate 34 fixed between the arms of the U-core 26so the lower face of the coil 32 is in the plane of the poles 28. Thecoil 32 provides the flux rotation signals. The signals from the coils30b and 32 are amplified by the head amplifier 25 before transmission tothe unit 18.

On the same former as the coil 32 there may also be wound a 200-turnrectangular coil whose windings are perpendicular to the plane of thefigure so the axis of the coil is parallel to the line between thecentres of the poles. This coil would provide flux leakage signals.Alternatively such a flux leakage coil may be provided somewhat furtherfrom the surface, for example above the support plate 34.

In operation of the system 10, the motor 14 is supported so the lowerend of the probe 12 is adjacent to the surface of the object 16 and thelongitudinal axis of the probe 12 is normal to the surface. Analternating current of the desired frequency and amplitude is suppliedto the drive coil 30a, so the magnetic field in the object 16 oscillatesabout zero with an amplitude much less than saturation. The probe 12 isfirst placed adjacent to a region of the object 12 where the stressesare negligible. The in-phase and quadrature components of the fluxlinkage signal (i.e. the component in phase with the drive current, andthe component at 90° to the drive current) received by the microcomputer20 are each backed off to zero, and the backing off values are thenfixed. During all subsequent measurements the flux linkage signalcomponents are backed off by these same amounts (i.e. subtracting asignal equal to the component observed when in a stress-free location).

Measurements can be taken by placing the probe 12 adjacent to a regionin which material properties such as stress are to be measured. Theorientation of the line joining the centres of the poles 28 (referred toas the orientation of the probe 12) is noted relative to a fixeddirection on the surface. The motor 14 is then energized to rotate theprobe 12, for example in a step-wise fashion 100 at a time through atotal angle of 360°. At each orientation of the probe 12 the quadratureflux rotation signal is measured, and the flux linkage components aremeasured (and backed off). These measurements may be made at differentfrequencies, for example with a drive current frequency of 68 Hz whenmeasuring flux rotation, and at a frequency of 300 Hz when measuringflux linkage. More typically the drive current frequency would be thesame when measuring both parameters.

It will be appreciated that the signal analysis procedure of theinvention is applicable with many different probes. The probe 12 mightfor example be modified by using a U-core 26 of a different materialsuch as silicon iron (which can provide higher magnetic fields), orindeed the drive coil might be air-cored.

The probe might be of a different shape or size, for example forinspecting surface stress in a small bearing it may be appropriate touse a probe of diameter as small as 3 mm, and to operate at a higherfrequency such as 100 kHz, while for inspecting internal stresses in alarge steel pipe it may be appropriate to use a probe of diameter say 75mm.

The flux rotation signals vary sinusoidally with probe orientation, sothe orientation at which they have their maxima and minima can bedetermined. The directions midway between these two orientations are thedirections of the principal stress axes. Measurements of the fluxrotation signals are therefore useful if the principal stress directionsare unknown. The flux linkage and flux leakage signals also varysinusoidally with probe orientation (in antiphase with each other) andthe values are observed at the principal stress directions. If theprincipal stress directions are already known, then the probe 12 mightinstead be merely oriented to those directions, and the flux linkagemeasurements made; no rotation of the probe 12 would be necessary.

The values of the stresses in the directions of the principal stressaxes can be determined from the experimental measurements of the fluxlinkage signals with the probe 12 oriented in those directions. Thisrequires calibration of the apparatus 10, taking measurements on asample of material of the same type as that of the object 16, whilesubjecting it to a variety of different stresses. This may be done witha rectangular strip sample in a test rig, measurements being made at thecentre of the sample where the principal stress direction is alignedwith the axis of the test rig. Referring to FIG. 3 this shows thebacked-off flux linkage in-phase and quadrature components obtained insuch a test rig, the measurements being made with a drive frequency of70 Hz, and the specimen being a steel bar. A first set of measurementswere made at progressively larger values of lift-off, L, but with nostress, S. This gives the lift-off line or contour A, the lift-offvarying between 0 and 220 μm. Similar lift-off lines A are obtained forother fixed values of stress, those for S=250 MPa tension andcompression being shown. Measurements were then made at a range ofdifferent fixed values of lift-off, L, with varying stresses, S (bothcompression and tension), providing the contours B.

It will be appreciated that the contours A are curved, and the contoursB are not orthogonal to the contours A, but that they intersect atangles that are constant along any one lift-off line A. Consequently itis only necessary to make calibration measurements sufficient to plotone such contour B and two or three such contours A, and the shapes ofthe other contours can be predicted. It has also been found that theintersection angles, and the curvature of the contours, arecharacteristic of the material.

After calibrating the probe 12 in this manner, measurements of stress orof lift-off can be readily made from observations of flux linkagesignals (resolved and backed off), as the contours enable the changesdue to lift-off to be readily distinguished from changes due to stress.Any particular position in the impedance plane (i.e. in the graph ofquadrature against in-phase components) corresponds to a particularvalue of stress and a particular value of lift-off. The mapping between(in-phase, quadrature) coordinates and (stress, lift-off) coordinatesmay be carried out graphically, referring to such contours, or bycalculation. For example if the flux linkage signal has the in-phase andquadrature components of the position marked X, this corresponds to alift-off of about 80 μm and a stress of about −125 MPa. Alternativelythe value X may be translated along a contour B of constant lift-off tofind the two components (at position R) for zero stress, and so to findthe lift-off. Similarly this value X may be translated (along the brokenline Y) along a contour A of constant stress to find the in-phase andquadrature components at position Z for zero lift-off.

Determination of the positions R and Z may be carried out graphically,as shown in the figure.

Another method is to deduce theoretical equations for how the signalwill vary with lift-off and material properties, taking into account thereluctance of the three parts of the magnetic circuit: the U-core 26,the air gaps, and the magnetic path through the object 16.

To provide an accurate fit to the experimental data, it would bedesirable also to take into account any losses in the core 26 andhysteresis losses in the object 16, and also flux leakage.Alternatively, simple polynomials may be used to model the experimentalmeasurements; although this approach provides little insight into thephysical phenomena it has been found to be the most accurate approach.This latter approach may be modified, based on the experimentalobservation that the intersection angles along any lift-off line and aset of stress lines remains constant.

The contours of constant stress, σ (referred to as lift-off lines) andcontours of constant lift-off, L (referred to as material or stresslines) constitute a non-linear two-dimensional matrix on the impedanceplane. The requirement is to determine for any subsequent impedancemeasurement the independent vector lengths on the impedance planecorresponding to a change in material and a change in lift-off. That isto say, the measurement, (x_(m),y_(m)) corresponding to the position Xmust be separated into the two specific components (x′_(σ),y′_(σ)) atposition Z, and (x′_(L),y′_(L)) at position R, where these twocomponents must lie on the zero lift-off stress line and the zero stresslift-off line respectively.

This analysis may be achieved in the following stages:

-   (A) General analysis for the lift-off/stress matrix measurements-   (1) The lift-off/stress matrix is parameterised using simple    polynomial equations. Typically this matrix may consist of between 3    and 5 lift-off lines and between 5 and 10 stress lines.-   (2) All the intercepts between the lift-off and stress lines are    then determined.-   (3) Cubic fits are carried out for both in-phase and quadratic    intercept values against the actual lift-off levels used when    measuring the original stress lines (provided there are at least 4    measured stress lines).-   (4) Stress lines can then be evaluated for any arbitrary lift-off    using the cubic fit coefficients (provided there are at least 3    measured lift-off lines).-   (B) Specific analysis for each subsequent measurement X-   (5) A new lift-off line is evaluated which passes exactly through    the measurement point.-   (6) The intersection, Z, of this new lift-off line with the zero    lift-off stress line is determined, (x′_(σ),y′_(σ)).-   (7) Find the stress line that passes exactly through the measurement    point.-   (8) The intersection, R, of this new stress line with the zero    stress lift-off line is determined, (x′_(L),y′_(L)).-   (9) Alternatively if stages (3) and (4) have been carried out then    the new stress line can be compared to the cubic fit coefficients to    determine the absolute lift-off level.

The above approach will now be described in greater detail, using thenumbering of the steps given above.

-   (1) All the lift-off (x_(L),y_(L)) and stress (x_(σ),y_(σ)) lines    are first fitted with quadratic equations:    y _(L)(σ)=a _(L)(σ)x _(L) ² +b _(L)(σ)x _(L) +c _(L)(σ)  (1)    (where the coefficients are different for each different value of    a), and    y _(σ)(L)=a _(σ)(L)x _(σ) ² +b _(σ)(L)x _(σ) +c _(σ)(L)  (2)    (where the coefficients are different for each different value of    lift-off)

It can be useful to evaluate the intersection angle between the lift-offand stress lines, y, at an arbitrary intersection position (x′,y′):$\begin{matrix}{{\tan\quad\gamma} = \frac{{2\left( {{a_{L}(\sigma)} - {a_{\sigma}(L)}} \right)x^{\prime}} + \left( {{b_{L}(\sigma)} - {b_{\sigma}(L)}} \right)}{1 + {\left( {{2{a_{\sigma}(L)}x^{\prime}} + {b_{\sigma}(L)}} \right)\left( {{2{a_{L}(\sigma)}x^{\prime}} + {b_{L}(\sigma)}} \right)}}} & (3)\end{matrix}$

For the lift-off lines we shall specifically identify the most extremecases that mark the edges of the matrix, for example those forcompressive and tensile stresses of 250 MPa in the figure.i y_(L)(σ_(c))=a _(L)(σ_(c))x _(L) ² +b _(L)(σ_(c))x _(L) +c_(L)(σ_(c))  (4)where σ_(c) is always the largest compression, andy _(L)(σ_(l))=a _(L)(σ_(l))x _(L) ² +b _(L)(σ_(l))x _(L) +c_(L)(σ_(l))  (5)where σ_(t) is the largest tensile stress (or at any rate the tensilestress giving the largest difference in signal from that at zerostress). Also the lift-off line measured at zero applied stress,y _(L)(0)=a _(L)(0)x _(L) ² +b _(L)(0)x _(L) +c _(L)(0)  (6)For the stress lines we shall identify the one at zero lift-off,y _(σ)(0)=a _(σ)(0)x _(σ) ² +b _(σ)(0)x _(σ) +c _(σ)(0)  (7)

-   (2) The intercepts between measured lift-off and stress lines are    determined by setting (x_(L),y_(L)) and (x_(σ),y_(σ)) equal to    (x_(σL),y_(σL)) in equations (1) and (2), and solving using the    fitted polynomial coefficients: $\begin{matrix}    {x_{\sigma\quad L} = \frac{\begin{matrix}    {\left( {{b_{\sigma}(L)} - {b_{L}(\sigma)}} \right) \pm} \\    \sqrt{\left( {{b_{L}(\sigma)} - {b_{\sigma}(L)}} \right)^{2} - {4\left( {{a_{\sigma}(L)} - {a_{L}(\sigma)}} \right)\left( {{c_{\sigma}(L)} - {c_{L}(\sigma)}} \right)}}    \end{matrix}}{2\left( {{a_{\sigma}(L)} - {a_{L}(\sigma)}} \right)}} & (8)    \end{matrix}$    then y_(σL) from (1) or (2). (9)    Care must be taken to select the solution of equation (8) which lies    within the bounds of the lift-off/stress matrix.-   (3) The intercepts, (x_(σL), y_(σL)), along each lift-off line are    fitted to cubic equations to parameterise the true lift-off values    using the 5 to 10 experimental lift-off values.    x _(σL) =p _(x)(σ)L ³ +q _(x)(σ)L ² +r _(x)(σ)L+s _(x)(σ)  (10)    y _(σL) =p _(y)(σ)L ³ +q _(y)(σ)L ² +r _(y)(σ)L+s _(y)(σ)  (11)    where the two sets of coefficients p, q and r for x and y are    numbers that are different for each value of stress (three in this    example).-   (4) A general complete set of stress lines can then be determined    using the coefficients in equations (10) and (11) for all lift-offs    within the measurement range and for small extrapolations beyond    that range. I.e.    -   evaluate general (x_(σL),y_(σL)) using equations (10) and (11)        for each measured lift-off line, then for each lift-off,    -   L=L′, collect all intercepts (x_(σL)′,y_(σL)′) and fit        quadratics to these as in equation (2) but now we have generated        stress lines for all specified lift-off values, these lines        making smooth transitions between the measured stress lines.

(B) Specific analysis for each measurement on component or plant.

-   (5) For a measurement (x_(m),y_(m)), the lift-off line which passes    directly through it can be expressed as (referring to equations    (4)-(6)): $\begin{matrix}    {y_{L} = \begin{matrix}    {{\left\lbrack {{a_{L}(0)} + {\Delta\left( {{a_{L}\left( \sigma_{t} \right)} - {a_{L}\left( \sigma_{c} \right)}} \right)}} \right\rbrack x_{L}^{2}} + {\quad\left\lbrack {{b_{L}(0)} + {\Delta\left( {{b_{L}\left( \sigma_{t} \right)} -} \right.}} \right.}} \\    {{\left. \left. {b_{L}\left( \sigma_{c} \right)} \right) \right\rbrack x_{L}} + \left\lbrack {{c_{L}(0)} + {\Delta\left( {{c_{L}\left( \sigma_{t} \right)} - {c_{L}\left( \sigma_{c} \right)}} \right)}} \right\rbrack}    \end{matrix}} & (12)    \end{matrix}$    where the parameter, Δ, represents the relative position across the    lift-off/stress matrix along the stress lines, and is given by:    $\begin{matrix}    {\Delta = \frac{y_{m} - {{a_{L}(0)}x_{m}^{2}} - {{b_{L}(0)}x_{m}} - {c_{L}(0)}}{\begin{matrix}    \left\lbrack {{\left( {{a_{L}\left( \sigma_{t} \right)} - {a_{L}\left( \sigma_{c} \right)}} \right)x_{m}^{2}} +} \right. \\    \left. {{\left( {{b_{L}\left( \sigma_{t} \right)} - {b_{L}\left( \sigma_{c} \right)}} \right)x_{m}} + \left( {{c_{L}\left( \sigma_{t} \right)} - {c_{L}\left( \sigma_{c} \right)}} \right)} \right\rbrack    \end{matrix}}} & (13)    \end{matrix}$    In equation (12) the expressions within square brackets are    numerical coefficients, say A₁, B₁ and C₁. This parameter Δ    effectively subdivides the area where there are contours, giving    equally spaced lift-off lines with a smooth variation, based on just    three measured lift-off lines. Note that this analysis does not use    the information determined in steps (2), (3), or (4) above. (6) The    intersection of the specific lift-off line (12) with the zero    lift-off stress line, (7), (x′_(σ),y′_(σ)), is given by:    $\begin{matrix}    {x_{\sigma}^{\prime} = \frac{\begin{matrix}    {\left( {{b_{\sigma}(0)} - B_{1}} \right) \pm} \\    \sqrt{\left( {B_{1} - {b_{\sigma}(0)}} \right)^{2} - {4\left( {{a_{\sigma}(0)} - A_{1}} \right)\left( {{c_{\sigma}(0)} - C_{1}} \right)}}    \end{matrix}}{2\left( {{a_{\sigma}(0)} - A_{1}} \right)}} & (14)    \end{matrix}$    then y′_(σ)from equation (7) by substitution. (15)

Here again care must be taken to select the correct solution.

(7-9) Find the stress line that passes exactly through the measurementpoint. In step (4) above, all general stress lines (and theircoefficients) were determined. Therefore it is only necessary todetermine which line includes the measurement point, (x_(m),y_(m)).There are several ways to do this, one being to determine the shortestdistance, D, from the point to each stress line:D(L)={square root}{square root over ((x _(m) −x* _(σ)(L)²⁺⁽ y _(m) −y*_(σ() L))²)}  (16)where the (x*_(σ),y*_(σ)) are the closest corresponding positions oneach stress line given by the solution to the simultaneous equations:$\begin{matrix}{{{y_{\sigma}^{*}(L)} - y_{m}} = \frac{x_{m} - {x_{\sigma}^{*}(L)}}{{2{a_{\sigma}(L)}{x_{\sigma}^{*}(L)}} + {b_{\sigma}(L)}}} & (17)\end{matrix}$  (2a _(σ)(L)x _(σ)*(L)+b _(σ)(L))(a _(σ)(L)x _(σ) ²*(L)+b_(σ)(L)x _(σ)*(L)+c _(σ)(L)−y _(m))+(x _(σ)*(L)−x _(m))=0  (18)

Thus for the range of L we find D(L)]_(min) and so determine theabsolute lift-off, L, for the measurement.

An advantage of this method, using stress lines determined using thecoefficients in equations (10) and (11) for all different values oflift-off (as outlined in step (4) above), is that it makes use of datafrom all the different experimentally-determined stress lines (of whichthere are seven shown in the figure, from L=0 to L=220 pm).

Alternatively if absolute lift-off value are not used and stages (3) and(4) have not been carried out, then by working with polynomialintersections in a manner analogous to that for the lift-off lines, theintersection point R can be obtained.

An alternative analysis approach takes into account the experimentalobservation of a constant intersection angle, γ, along each lift-offline, in determining the equations of the lift-off lines and stresslines. The result is more robust to extrapolation beyond the calibratedlift-off range and requires fewer measurements of stress and lift-offlines to construct.

The measured lift-off lines are fitted with simple quadratic equationsas described by equations (4) to (7). These are typically taken so as torepresent the two extreme values of stress and the zero stress lift-offline respectively (three lines, although more can be used). Only asingle stress line needs to be measured, and this can be at anylift-off, Lo, equation (2). The intersection angles, γ(σ), of thelift-off lines along the stress line are evaluated from equation (3).Let the tangent angle to any lift-off line be a_(L)(σ) and that to thestress line be B_(σ)(L₀). Then:tan a _(L)(σ)=2a _(L)(σ)x _(L) +b _(L)(σ)  (19)tan β_(σ)(L ₀)=tan(a _(L)(σ)−γ(σ)=tan[tan⁻¹(2a _(L)(σ)x _(L) +b_(L)(σ))−γ(σ)]=2a _(σ)(L₀)x _(L) +b _(σ)(L ₀)  (20)Expanding the more complicated tangent in equation (20): $\begin{matrix}{\frac{{2{a_{L}(\sigma)}x_{L}} + {b_{L}(\sigma)} - {\tan\quad{\gamma(\sigma)}}}{1 + {\left( {{2{a_{L}(\sigma)}x_{L}} + {b_{L}(\sigma)}} \right)\tan\quad{\gamma(\sigma)}}} = {{2{a_{\sigma}\left( L_{0} \right)}x_{L}} + {b_{\sigma}\left( L_{0} \right)}}} & (21)\end{matrix}$

In equation (21) all the polynomial coefficients and the gamma anglesare already known. However we know that γ(σ) is independent of thelift-off, and so we can generalise equation (21) to all lift-off values,L: $\begin{matrix}{\frac{{2{a_{L}(\sigma)}x_{L}} + {b_{L}(\sigma)} - {\tan\quad{\gamma(\sigma)}}}{1 + {\left( {{2{a_{L}(\sigma)}x_{L}} + {b_{L}(\sigma)}} \right)\tan\quad{\gamma(\sigma)}}} = {{2{a_{\sigma}(L)}x_{L}} + {b_{\sigma}(L)}}} & (22)\end{matrix}$where now the stress line coefficients and the intercept between thisline and any lift-off line (RHS of equation (22)) must be established.

To do this we simply solve the simultaneous equations (22) usingcoefficients from 2 or 3 lift-off lines together with equations (2),(4), (5) and (6) to determine the stress line coefficients and interceptpositions. Note that any x position, x_(L), can be chosen on a lift-offline and the corresponding y position and stress line coefficientsdetermined for that case.

Hence the position Z can be determined, eliminating the effect of anylift-off, and this can be related to stress. The value of stress foundin this way is, it will be appreciated, the uniaxial stress that wouldprovide that value of the flux linkage signal. If the stresses areactually biaxial, then a further calibration must be carried out with across-shaped sample in a test rig, flux linkage signal measurementsbeing made at the centre of the sample where the principal stressdirections are aligned with the axes of the test rig. Hence a graph ormap may be obtained for a range of values of stress on one principalaxis (say the x-axis) and for a range of values of stress in the otherprincipal axis (say the y-axis), with contours each of which shows thevalues of biaxial stress that give a particular value of apparentuniaxial stress along the x-axis; and a similar map may be obtained withcontours showing values of biaxial stress that give a particular valueof apparent uniaxial stress along the y-axis. Hence from measurements ofapparent uniaxial stress along the two principal stress axes obtained asdescribed earlier, the biaxial stress can be determined.

It will again be appreciated that the biaxial stress may be determinedeither graphically or by calculation in this way. Apparent values ofuniaxial stress (in MPa) may be used for this purpose, or alternativelythe numerical value of the flux linkage signal (in mV), either thein-phase or quadrature value, obtained by eliminating the effect oflift-off as described in relation to FIG. 3, may be used.

For some materials, such as hard steel, the biaxial stress contours fordifferent values of apparent uniaxial stress (or of flux linkage signal)along the x-axis, and those along the y-axis, intersect at large anglesover the entire biaxial stress plane. Finding the intersection of twosuch contours, and so the true value of the biaxial stress, can be doneeasily. However for mild steel the two sets of contours (particularly inthe tensile/tensile quadrant) are almost parallel, so it is verydifficult to locate their intersection. In this case measurements with adifferent stress-dependent variable are helpful, as they can provide athird set of contours; the flux rotation signals are desirably used forthis purpose.

It will be appreciated that the invention enables lift-off to bedistinguished from changes in material properties (such as stress), overa wide range of values of both lift-off and material properties, as itdoes not require an assumption that the changes are linear in theimpedance plane. The probe described above is just one type ofelectromagnetic probe for which the invention is applicable. Theelectromagnet means may include a ferromagnetic core on which a drivecoil is wound, such as a C-core or a rod, or may be air cored. Thesensor may be a sensing coil wound on the same core, or a sensing coilpositioned near to a pole, or may indeed be the same coil as the drivecoil. There may also be several sensor coils, for example in an array.

Measurements of lift-off require at least a region of surface on whichthere is no coating, either on the object 16 or on a test piece of thesame material, so that a measurement at zero lift-off can be made.Subsequently the thickness of any coating of non-ferromagnetic materialcan be measured, as it is equal to the lift-off if the probe is incontact with the coating. As a rule, a probe smaller than that describedabove would be more sensitive to lift-off and less sensitive to materialproperties, so that if the primary intention is to measure lift-off thena probe of external diameter say 8 mm or 4 mm might be used. The shapeof a surface, for example any undulations in the upper surface of a railhead, may therefore be measured in a non-contact fashion by scanning asingle probe over the surface and monitoring the changes of lift-offdetermined as described above; alternatively an array of probes might beused for this purpose, and might also be scanned in this fashion. Whentaking measurements on a surface that is curved, it is desirable for theface of the probe to have a similar curvature.

It will thus be appreciated that the present invention is particularlysuited to the determination of stress in a ferromagnetic material, aschanges in stress have a large effect on the relative permeability,μ_(r), while having a negligible effect on conductivity. A singlespecimen can be used for generating the signal/stress data and thestress/lift-off data, by subjecting the specimen to a range of differentuniaxial or biaxial stresses. The resulting data can subsequently beused in taking measurements on many different objects of the same typeof material. In general there is a need to determine three independentparameters—the direction of the principal axes, and the stresses alongeach principal axis. It is hence desirable, in general, to takemeasurements with different orientations of the probe.

1-5. (Cancelled)
 6. A method as claimed in claim 10 wherein theelectromagnet means comprises an electromagnetic core and two spacedapart electromagnetic poles), and the magnetic sensor is arranged tosense the reluctance of that part of the magnetic circuit between thepoles of the electromagnet means.
 7. (Cancelled)
 8. A method as claimedin claim 10 wherein the contours are deduced in the first calibrationand are represented by polynomials in performing the mapping.
 9. Amethod as claimed in claim 8 wherein the contours are presented bypolynomials in performing the mapping, the constant intersection anglealong a contour representing signal variation with lift-off being takeninto account in generating the polynomials.
 10. A method for measuringbiaxial stresses in an object of ferromagnetic material, the methodusing at least one probe the or each probe comprising an electromagnetmeans and a magnetic sensor arranged to sense a magnetic field due tothe electromagnet means, the method comprising arranging the probeadjacent to an object, and activating the electromagnet means so as togenerate an alternating magnetic field in the electromagnet means andconsequently in the object, the field alternating at a frequencyselected so as to provide a desired depth of penetration below thesurface of the object; detecting signals from the magnetic sensor andresolving the signals into a first component and a second componentwhich are orthogonal in phase, and mapping the inphase and quadraturecomponents directly into apparent stress and lift-off components bymeans of a first calibration, using a sample of the ferromagneticmaterial and subjecting it to a range of different uniaxial stresses andwith the probe at a range of different values of lift-off so as todeduce the corresponding contours in the impedance plane; arranging theprobe adjacent to the ferromagnetic object, turning the probe todifferent orientations and detecting signals from the magnetic sensor atleast when the probe is oriented with the two principal stress axes, andfrom the contours in the impedance plane determined in the firstcalibration deducing the corresponding two values of apparent uniaxialstress along the two principal stress axes; and, by means of a secondcalibration with a sample of the ferromagnetic material subjected to arange of different biaxial stresses, deducing the true biaxial stressfrom the two values of. apparent uniaxial stress.
 11. A method asclaimed in claim 10 wherein the second calibration entails measuring aset of contours each of which shows the values of biaxial stresses thatgive a particular value of apparent uniaxial stress along one principalstress axis, and a second set of. contours each of which shows thevalues of biaxial stresses that give a particular value of apparentuniaxial stress along the other principal stress axis.